openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /PolarAlgebraSeparation.lean
| import OAI.Analysis.IrrationalRotation.Spectral.ProductPolarComplete | |
| import OAI.Analysis.IrrationalRotation.Product.ProductHilbert | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| theorem finitePolarAlgebra_le_fullPolarAlgebra | |
| (U B : H →L[ℂ] H) (F : Finset ℤ) : | |
| finitePolarAlgebra U B F ≤ fullPolarAlgebra U B := by | |
| apply vonNeumannGenerated_le | |
| rintro A ⟨n, _hn, rfl⟩ | |
| exact subset_vonNeumannGenerated _ ⟨n, rfl⟩ | |
| theorem not_isDiffuse_of_isAtomic [Nontrivial H] (M : VonNeumannAlgebra H) | |
| (hat : IsAtomicVonNeumannAlgebra M) : ¬ IsDiffuseVonNeumannAlgebra M := by | |
| intro hdiff | |
| obtain ⟨p, hp, _⟩ := hat 1 (one_mem M) (IsStarProjection.one _) one_ne_zero | |
| exact hdiff p hp | |
| theorem ProductPolarStructure.finitePolarAlgebra_lt_fullPolarAlgebra | |
| [Nontrivial H] {U T : H →L[ℂ] H} | |
| (h : ProductPolarStructure U T) (F : Finset ℤ) : | |
| finitePolarAlgebra U (operatorModulus T) F < | |
| fullPolarAlgebra U (operatorModulus T) := by | |
| apply lt_of_le_of_ne (finitePolarAlgebra_le_fullPolarAlgebra _ _ _) | |
| intro heq | |
| have hat := (h.2.2.2.2.2.1 F).2 | |
| rw [heq] at hat | |
| exact not_isDiffuse_of_isAtomic _ hat h.2.2.2.2.2.2.2 | |
| namespace ProductModel | |
| theorem finitePolarAlgebra_lt_fullPolarAlgebra_of_logIntegral | |
| (θ c : ℕ → ℝ) (hadm : AdmissibleAnglesAndCosts θ c) | |
| (hlog : HasNegInfiniteLogIntegral baseMeasure (weight θ c)) (F : Finset ℤ) : | |
| finitePolarAlgebra shiftOperator (operatorModulus (weightedOperator θ c)) F < | |
| fullPolarAlgebra shiftOperator (operatorModulus (weightedOperator θ c)) := | |
| ProductPolarStructure.finitePolarAlgebra_lt_fullPolarAlgebra | |
| (productPolarStructure_of_negInfiniteLogIntegral θ c hadm hlog) F | |
| end ProductModel | |
| end Invariant233 | |
| end | |
| end OAI | |